Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
418 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2177 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#2176 | C2 | semidirect |
| 4. #13657 | C2 | direct (N × H) |
| 5. C2×#1426 | C2 | semidirect |
| 6. C2×#1420 | C2 | semidirect |
| 8. C2×#1398 | C2 | semidirect |
| 13. #2264 | C4 | semidirect |
| 15. C2×Q64 | C4 | semidirect |
| 16. C2×SD64 | C4 | semidirect |
| 22. #2176 | C22 | semidirect |
| 28. #1426 | C22 | semidirect |
| 29. #1420 | C22 | semidirect |
| 31. #1398 | C22 | semidirect |
| 60. Q64 | C2×C4 | semidirect |
| 61. SD64 | C2×C4 | semidirect |
| 237. C2 | #13657 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2264 | 2 |
| C2×#2179 | 2 |
| C2×#2176 | 2 |
| #13657 | 2 |
| C2×#1426 | 2 |
| C2×#1420 | 2 |
| C2×#1418 | 2 |
| C2×#1398 | 2 |